STEADY-STATE ELECTRICAL-CONDUCTION IN THE PERIODIC LORENTZ GAS

We study nonequilibrium steady states in the Lorentz gas of periodic s catterers when an electric external field is applied and the particle kinetic energy is held fixed by a ''thermostat'' constructed according to Gauss' principle of least constraint (a model problem previously s tudied numerically by Moran and Hoover). The resulting dynamics is rev ersible and deterministic, but does not preserve Liouville measure. Fo r a sufficiently small field, we prove the following results: (1) exis tence of a unique stationary, ergodic measure obtained by forward evol ution of initial absolutely continuous distributions, for which the Pe sin entropy formula and Young's expression for the fractal dimension a re valid; (2) exact identity of the steady-state thermodynamic entropy production, the asymptotic decay of the Gibbs entropy for the time-ev olved distribution, and minus the sum of the Lyapunov exponents; (3) a n explicit expression for the full nonlinear current response (Kawasak i formula); and (4) validity of linear response theory and Ohm's trans port law, including the Einstein relation between conductivity and dif fusion matrices. Results (2) and (4) yield also a direct relation betw een Lyapunov exponents and zero-field transport ( = diffusion) coeffic ients. Although we restrict ourselves here to dimension d = 2, the res ults carry over to higher dimensions and to some other physical situat ions: e.g. with additional external magnetic fields. The proofs use a well-developed theory of small perturbations of hyperbolic dynamical s ystems and the method of Markov sieves, an approximation of Markov par titions.